Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994.Includes bibliographical references (leaf 17).by Ana M.L.G. Canas da Silva.M.S

Silva, Ana M. L. G. Canas da

DSpace@MIT

Atiyah-Bott Theory for Orbifoldsand Dedekind SumsbyAna M. L. G. Canas da SilvaLicenciatura em Matemitica Aplicada e ComputaaoInstituto Superior T6cnico, Universidade Tecnica de Lisboa, 1990Submitted to the Department of Mathematicsin Partial Fulfillment of the Requirements for the Degree ofMaster of Science in Mathematicsat theMassachusetts Institute of TechnologyFebruary 1994@1994 Ana M. L. G. Canas da Silva. All rights reserved.The author hereby grants to MIT permission to reproduce and to distribute publiclypaper and electronic copies of this thesis document in whole or in part.Signature of Author , .. . v ...Department of MathematicsJanuary 10, 1994Certified by -Victor Guillemin, Professor of MathematicsThesis SupervisorAccepted by A 6' , / {"David Vogan, ChairmanDepartmental Committee on Graduate StudiesAPR 2 5 1994Lgi ScienceAtiyah-Bott Theory for Orbifoldsand Dedekind SumsbyAna M. L. G. Canas da SilvaSubmitted to the Department of Mathematicson February 1, 1994 in Partial Fulfillmentof the Requirements for the Degree ofMaster of Science in MathematicsABSTRACTThis paper shows how to deduce the reciprocity laws of Dedekind and Rademacher,as well as n-dimensional generalizations of these, from the Atiyah-Bott formula, byapplying it to appropriate elliptic complexes on a "twisted" projective space. Thistwisted projective space is obtained by taking the quotient of C" - 0 by the actionp(w)(Zl1,. .- ,Zn) = (Wql ,qZn), E C, qi E Z+ ,where the qi's are mutually prime. Since this is not a manifold, it is necessary toadapt Atiyah-Bott to the setting of orbifolds.Thesis Supervisor: Victor GuilleminTitle: Professor of Mathematics2Contents1 Fixed point formula for orbifolds 51.1 The case of good orbifolds ........................ 51.2 The case of general orbifolds ....................... 62 Application to a twisted projective space 82.1 General formula .............................. 82.2 The limit case ............................... 102.3 The case n = 3 and reciprocity laws .................. . 112.4 Generalized Dedekind sums ....................... 132.5 Counting lattice points . . . . . . . . . . . . . . . . . ....... 153IntroductionLet Y be the "twisted" projective space obtained by taking the quotient of C ' - 0by the actionp()(Zl,... ,zn) = (ql'z 1,. .,q"z,), E C,qi E Z+ ,where the qi's are mutually prime. We will show in this paper how to deduce thereciprocity laws of Dedekind and Rademacher, as well as n-dimensional generaliza-tions of these formulas, from the Atiyah-Bott formula by applying it to appropriateelliptic complexes on Y. Since the twisted projective space, Y, is not a manifold,this will require our adapting Atiyah-Bott to the setting of orbifolds. The versionof Atiyah-Bott needed for our purposes is described in section 1 and the numbertheoretic applications of it, mentioned above, are discussed in section 2.41 Fixed point formula for orbifolds1.1 The case of good orbifoldsLet X be a compact complex manifold of complex dimension n, G a finite group actingon X with action r: G x X - X. The quotient space Y = X/G is consequently agood orbifold.Define the Dolbeault cohomology of Y to be the G-invariant cohomology of X,H'(Y) = HG(X), where HG(X) are the G-invariant subspaces of Hi(Y),i = 1,...,n.A holomorphic G-equivariant function f: X - X induces a quotient map f: Y - Yand f: Hb(X) Hb(X) from the pull-back on G-invariant forms.We will define the Lefschetz number of the mapping f to benL(f) = Z(-1)t trace (ft: Hb(X) -- H(X)).i=1We will need the following elementary result:Theorem. 1.1 Let V be a vector space and p : G - Aut(V) a representation of afinite group G on V. IfL : V - V is a G-equivariant linear map, then1trace (L: VG -- VG) = II E trace (p o L) : V -- V),where VG is the space of G-fixed vectors in V.By the above Theorem 1.1, we haveL(f) = ' I E trace ((Tgr of) : Hi(X) --+ H(X)).i-1 GI EGSupposing, in addition, that f: Y -+ Y has only non-degenerate isolated fixed points,or equivalently, that rg o f has only non-degenerate isolated fixed points for all g E G,we can compute-(-1)'trace((rgof) : H'(X) - H'(X))= E sgndet(l-d(rgof)p) (1)i=1 {pl(rgo)(p)=p}by the standard Lefschetz fixed point theorem [GP].Theorem. 1.2 Under the above conditions we have:L(f) C= - C E sgn det (1 -d(r, o f)p).gIEG {pl(-gof)(p)=p}1.2 The case of general orbifoldsIn order to write formula (1) in a form which makes sense for general orbifolds Y thatare not globally quotients of the form X/G (X a manifold, G a finite group), let usdetermine the actual contribution of a fixed point q of f Y - Y. Still assumingY = X/G, let Pi,P2, ,Pk be the pre-images of q in X. Replacing, if necessary, fby r o f for some g E G, we can assume f(pl) = pi. Let Gpi be the stabilizer groupof pi in G. Thus, the contribution of q to the Lefschetz number is:1 E sgn det (1 - d(r9 o f)p,)=l 1 {gGl(rgol)(Pi)=pi}orG1 i, sgn det (1 - d(r, o f)p,)I i=1 9EGpisince f is G-equivariant. In fact,Z sgn det (1 - d(rg o f)p) =  sgn det (1 - d(rg o f)p,iEGpi gEGplbecause the Gp,, are conjugate and f is G-equivariant, i.e.d(rTg O Tg rg. 1 o f)rg;(p) = d(rg O f)p.6Therefore, the contribution of q to L(f) is1 k * sgn det (1-d(rgof)pl)I GI gEGpor1 ,Z sgn det (1 -d(T o f)p,).IGP1 |IEGplThis motivates the following result (which we will give a proof of elsewhere):Let : Y Y be a holomorphic function from a compact complex orbifold Y to itself,having only non-degenerate isolated fixed points ql,...,q,,,. Define the Lefschetznumber of f to benL(f) = Z(-1)itrace (f : H'(Y) _-_ H(Y)).i=lTaking orbifold charts around each of the qi's, for a neighborhood Y of qi, there are:X and G such that Yi = X/G,a pre-image pi of qi,an isotropy group Gi, anda locally well-defined lift fi of f.Claim: We havem 1L(f) = sgn det (1 - d(g o fi)pi)i=1 7G I EGireducing again a global topological invariant to a finite number of local differentialcomputations.Remark: If L -+ G is a G-invariant holomorphic line bundle and H'(X,L) thecohomology groups obtained by tensoring the Dolbeault complex with L, we cancompute the alternating sum of the traces of fl on Hb(X, L) by a sum over the fixedpoints of I: Y -- Y of the terms1 trace (g o fi Lp ) Lp)Gi gEG, det (1 - d(rg o fi)p,) [AB.72 Application to a twisted projective space2.1 General formulaTake Y to be the orbifold obtained by dividing Cn - 0 by the group C* where C*acts byP(,)(Z,,.· ., ,n) = (1 Z1 ,. . W. Z,), qi E Z+ .Assuming ql, ... , qn mutually prime, the orbifold Y is non-singular except at the npoints:[1,0,...,0], [01,0...,0], .... [0,...,0, 1]which have stabilizers Z/ql,..., Z/q, respectively, and thus may be singular. (Noticethat when qi = 1, the corresponding point is non-singular.)The standard diagonal action of S' on C - 0,ft(Zi,. . .,Zn) = (e27rit1,.l . e2 ritZn)induces an action ft on the orbifold Y (since it commutes with p). As long as t Z 0,its fixed points are only[1,0,...,0], [0,1,0,...,0],..., [0,...,0,1].Consider the holomorphic line bundle L over Y associated with the representation: C* #-- Aut(C), Y(W)c = WdC,i.e., L = [(Cn -0) x C]/{[z,-7(w)c] [p(w)z,c],w E C}.In order for L to be well-defined on Y, the conditionqi1 d,i = 1,..., n, or equivalently, ql . qn Idis required. We will write d = . ql ... qn.8PROOF. The projection of the hyperplane z, = 1 of C - 0 on Y contains onlyone of the singular points, namely [0,..., 0, 1]. The subgroup of C' which fixes thiscross-section is the group of q roots of unity that acts as p(w)(z 1 ,...,z,._.,l) =(wQ zl,. . .wn-1 Zn-1l,1) , while y(w)c = wdc on the fiber of L. We have[(0,..., 0, 1), y()c] [p(w)(O,..., 0,1), c]It II[(0,..., , O1), wdc] [(0,...,0, 1), c]hence, in order for L to be well-defined at [0,..., 0, 1] we need q,,d.Similarly for the other singular points. Q.E.D.On the cross-section Zn = 1, rq = p(e2rq ) acts byTq(21***Z- 1e) =1 (e27riqn-1q .. e2r i qqZn _lwhereasft(Tz,.. z =1) (e2 itz... ,e2ritz 1n, e2 rit) A-, (e2nit(l )Zl ,. en n,2it(l- Z- ).ft(Z , Zn-1, 1) = (e (e2it( e L We define an action of S1 on L induced by letting S1 act trivially on the second factorof (Cn - O0) x C:et[(0..., 0,0,1), = [(0,..., 0, e2 ), c] [(0, ... .0,1), e2Rtc],so the action of e2"'it E S1 on the fiber of L above [0,..., 0, 1] is given by multiplication27rit dby e q.Interpreting these results in terms of the lift to the smooth Cn-l covering of thiscross-section (which roughly amounts to ignoring the last coordinate Zn when it's 1),we conclude that= multiplication by e2 rit Lo...o) L(o...o)Tq a (ft)n = (ft)n = multiplication by e q : L(o.,0) - L(o,...,o)92iqd( (f ) 0) d' 1 diae2rit(l)zl, . I et(-1)Zn-d(qo(ft))(o...o) = diag( , , e 9 )iag(e q , t nZ1Summing over the qn-th roots of unity, w = e qn, q = 0, 1,..., qn-1, the contributionof [0,... ,0, 1] to the Lefschetz number of ft is1 q-1 e27ritdqn q=0 rIzmin -e2=ri(l1-2m )t 2 eriq=O Hmn( 1 - e- aqa e ,n )Similar computations yield similar results for the other fixed points. Adding all thesecontributions up we finally get for the global Lefschetz number of it:-n 1qr 2rit dL(ft) = 9r1 1 (2)r=l qr q=O nImr(1 _-2ai(l-qm)t . e (2)On the other hand, the Lefschetz number of ft was defined to ben-1L(ft) = Z(-l )'tace(] : H'(Y, L) - H'(Y, L)).i=lWe assume Hi(Y, L) = 0 for i > 0. As for H°(Y, L) this is the global holomorphicsections of L and these are just the monomials on C" z, ... z n which transformunder the action of C* according to the law(l Z)ml · · · (Wqn n)mn = wdml . . Zmnand hence qlml+. . .+qnm = d. Thus the dimension of H°(Y, L) is the number # ofintegerlatticepoints (ml,...,m,) satisfying qlml+...+qnmn = d, ml,... ,m, > 0.We will compute this dimension in the next section by studying the limit of (2) ast -- 0.2.2 The limit caseAlthough our formula doesn't hold for t = 0 since fo leaves all points fixed, we cancompute its limit as t --. 0. Notice that the dimension of H°(Y, L) is independent of t.10So, when t --+ 0,n q19~~r-l 2rit #= limZ- z et-r= q q0 =mr(1 - e'i(- I )t e 'i )n 1 qr-1 1= Ez-Er=l qr =l rHmnr(l - 2riq ) +e qr )n 1e27rit dli m r( e t-owhere the last limit can be computed writing a Laurent series for each summand:an-l,_ al,1tn- ltr + + ao, + .. - ..tn-I tAs t -, 0 the sums of the negative terms in these series must cancel and we end upwith• (ao,r +r=l1 qr-1qr q=l rlmr(1 - e2 ri a)as the number of non-negative integral solutions of the equationqlml + ... + qnmn = d.Now we can writen 1 qr-=r=1 qr =11 1Hm r(1 - e2' i ) r= q' r =l,, l1fjmr(j _-and relate this to generalized Dedekind sums according to [HZ] (see section 2.4).2.3 The case n = 3 and reciprocity lawsFor n = 3 (the first interesting case), formula (3) reads:#{(ml, m 2, m3) E Z3 lqlml + q2m2 + q3 m3 = d, ml, m 2, ms 3 0} =3 1 qr - 1E 27riUmr= q 1 lnmr(1 - e qr )A_ e + lim 1 e2ri(1 t 't-0 r=l qr In,(l - e2i( qr )B11(3)ILet's deal with each of these terms A and B in turn.A:We can write1r 7 9r=1,1 701 imr( - r1 Qm)Setting q3 - kl q2 mod ql, q - k2q3 mod q2, q2 = k3 q1 mod q3, we findA = Z - 1A = -- E (1 - 77)(1 - -) r~l qr 779r=11r01 r l 4q, s(krq,,)r1lby the definition of s(k, q,) according to [RG, p.151. But by the Rademacher reci-procity law [HZ, p. 9 6],3~~~ q 2 +2 + 2 1] s(kr,qr) = I q+: + q2 ' 1r= 12 qlq2q3 4When q3 = 1 we can take kl = q2, k2 = q, k3 = 0 and the formula reduces to3 112Z s(k, qr) = s(q2, q1) + s(ql, q2) = -r=l1which is just the Dedekind reciprocity law [RG].B:1-q2+ qlqq+ql q2 q1Each summand in B is of the form -qrterm in the Laurent expansion isao = -4qr,I w2 wleWt(1--ewlt)(1-ew2 tlw 1 W 2 1 W12 2 2 w lw 2 12 w2) for which the constant1w212wl Therefore,3 1/1B= Eao,r =--+r=1 4q! q21+1)1 12+ (qi + q2 + q3) + jq1l2q3 +2 21 q~2 4 + +212 q+q+q12 qlq~q3Next we should compute the left-hand side to see if it agrees. We have# {(ml, m 2, m3 ) E Z3Jqimi + q2mn2 + qsm3 = d = lqlqq3, in, 2, m 3 0}1213A=-rlIq q2=- #{(mi, m 2) E Z2 [Iqml + q2m 2 = (lqtq2 - m3 )q3, mI,m 2 > O}m3 =0+ 1 -6(m3) ,where[x] denotes the greatest integer not exceeding x,e(m 3) = 0 or 1, with e(m3) = 0 whenever m 3 is a multiple of ql or q2, ande(m 3 ) + e(lqlq2 - m3) = 1 when m3 is neither a multiple of q, nor of q2.Therefore,= I#{integers2 in [0, Iqjq2] neither multiples of ql, nor of q2} - (q-1)(q2-1)2Also, since(p - 1)(q + 12we getlql q2m3 =0122for p, q mutually prime,* qlq2q3 + (q3 -qlq2 + 1)-We conclude that#{(ml, m 2 , m3) E Z3 lqlml + q2m2 + q3m3 = d = qlq2q3, i, m, 2, m 3 > 0}12 1= - qlq2q3 + (ql + q2 + q3) + 12 2and, hence, in this case (3) is equivalent to Rademacher reciprocity law.2.4 Generalized Dedekind sumsWhen = 0, i.e., d = 0 and the line bundle L is trivial, formula (3) reduces ton 1 q r - 1 1r=l qr =l lmr((1 -e r )rI 1+lt-Or=l qr lmr(1 - e 2 rr -(1q)t13lql q2E (m3)m3 =01qj q (Iql2 - 3)q3M3=0 q, q2k= 1 JI(lqlq2 - 3)q3r 1 1r=1 qr =j7 VImr(1 -lrm)= 11 -limE - 1 rst- r= qr mpr(1 -- 92r(Iq )where the last limit can be evaluated by the Laurent series argument. Letting6(qr; q, i r) = , 7 Er1,7 ~  17qr=l 77, 1 lmor( - -qm)n 1an(ql,..., qn) = -n(qr; q, i r),r=l qrwhen n = 2, 3, 4, 5 we explicitly find the following generalized reciprocity laws.Cn(ql,. . .,qn)1 - -+- 2 q, q2j4 (q1- 1 18 qiQ11 1 \ 1 q + q2 + q3+-- +1)-1 q~+q~+q3q2 q3 12 qlq2q31 1 1+-+-+-q2 q3 q41 (ql+ q2 + ql + q3 + ql + q4 q2 + q324 q3q4 q2q4 q2q3 qlq 41 1 1 1 q qq- 4-8 qlq 2 q3q4q5 ij<k i____ ~ s~a~s;~T<;61 1 1144 qlq2q3q4q5 i<3 * 720i<) 2q2 + q4qlq3q3 + q41qlq 2 q3 q4 q5 Remark: It is possible to write the limit term (corresponding to B in section 2.3) interms of Bernoulli numbers Bn defined byBnt = tE n! et -1 'n=01 1Bo = 1, B =2 3...This is why the coefficients in the final expressions for the ac's resemble products ofBernoulli numbers.14(4)n =2n =3n=4n=5On the other hand, [HZ, p.100-101] gives results for generalized Dedekind sums oftype 6, for n odd, namely1 kq, l-l (  l.l. . qn)E 1E r icot -1- (5)r=l qr k=l m1  qr ql ... qnwhere In-l is a certain polynomial in n variables which is symmetric in its variables,even in each variable, and homogeneous of degree n - 1. Formula (5) is related to theprevious 6,'s and an's by1 rqr - 1 7rkq, 1 ____*= qr k=l mgr qr r=1 qr rqr=l,1 *l mor m - 1n 1 n-l (-2)ir=1 Q 77qr=1,771 j=0 IC{1l...n}\r,#I=j riI(1- 711 )n-1 n 1-2)j 6j+l(q; q, i E I)j=O r=l qr IC{(1...n\r,#I=jn= (-2) j -1 E aj(qi,i E J).j=l JC{1...n},#J=jWhen n = 3, 513 1 7 512(ql, q2, q3) = q /4(ql q2 q4 q5) q -90 qi=1 i=and it is easily seen that (5) is in agreement with our results for a,, n = 2, 3, 4, 5. Insome sense (4) extends (5) to the case of n even.2.5 Counting lattice pointsConsidering again a general line bundle (i.e., arbitrary d, or 1), we see that formula(3) provides an expression for the number # = #n(ql,. . , q,) of integer lattice points(mi, ..., mn) satisfying qlml + ... + qnM, = d, ml,..., mn > 0, namelyn qr-1 n 2irit #n(ql,..., q)= E' E - + lim1 e g'i(6)r=l qr q=1  m9r(r t- 1 Q l mmr(1 _ e2qi( 9 )t(An Bn15As in the case d = 1 = 0 (see formula (4)),n 1 qr-1 1r=l qr q=1 -mi#r(1 - e 21qr)n 1 1-lim 1rt-0 _ .26q, Hm1rO - e -2and thus both An and Bn can be computed from the Laurent series argument. Forn 5 we get the following results.#1 = 1#2 = 1+1#3= qlq2q3 + (q + q2 + q3) + 12 21 3 12#4 = ( 4 qlq2q3q4(ql + q2 + q3 + q4)6 4+ q+ q + q32 + q2 + 3qlq2± 3qq3 + 3q1q412+ 3q2q3 + 3q2q4 + 3qq4 ) + 1(II qj) ( q)#5 = (I + 24 12+ 12 (r qi) q22~4 S +3* E qiqj)i<jI24 qqi + 3 qiqjqk +1ikWorking out #n(ql,. .. , q,) directly for each n, by decomposing into sums generalizingthe procedure in section 2.2, e.g.lqlq2q3q4#4 = #{qlml + q2m2 = x} . #{q3m3 + q4m4 = lqlq2q3q4 - },X=Oand equating similar powers of I in (6), we can gradually find many other interestingformulas.We conclude that it is easy to deduce Number Theory results from Atiyah-Bottadapted for orbifolds, by applying it to specific examples.16References[AB] M. Atiyah and R. Bott, Notes on the Lefschetz Fixed Point Theorem for EllipticComplexes, Harvard University, Cambridge, 1964.[GP] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, EnglewoodCliffs, 1974.[HZ] F. Hirzebruch and D. Zagier, The Atiyah-Singer Theorem and Elementary Num-ber Theory, Publish or Parish, Inc., Boston, 1974.[RG] H. Rademacher and E. Grosswald, Dedekind Sums, The Carus MathematicalMonographs 16, The Mathematical Association of America, Washington, 1972.17

Atiya-Bott theory for orbifolds and Dedkind sums

http://dspace.mit.edu/bitstream/1721.1/28053/2/30952342-MIT.pdf

Atiya-Bott theory for orbifolds and Dedkind sums

Abstract

Similar works

Full text

Available Versions

DSpace@MIT